Hidden Triples in Sudoku: The Complete Guide

Hidden Triples in Sudoku: The Complete Guide

What is a Hidden Triple?

A hidden triple occurs when three candidates appear in only three cells within a unit (row, column, or box), but those cells also contain other candidates that "hide" the triple. Once you identify the three numbers that form the triple, you can eliminate all other candidates from those three cells.

Example:

In Box 5 (center box), candidates 2, 5, and 8 appear in only three specific cells:

  • Cell A: {2, 3, 5, 7, 8}
  • Cell B: {2, 4, 5, 6, 8}
  • Cell C: {2, 5, 8, 9}

No other cells in Box 5 contain 2, 5, or 8. This means these three numbers must occupy these three cells. Eliminate 3, 7 from Cell A; eliminate 4, 6 from Cell B; eliminate 9 from Cell C, leaving {2, 5, 8} in each cell.

Why Hidden Triples Matter

  • They unlock progress in crowded units where other techniques fail.
  • They simplify complex cells with many candidates.
  • They often reveal naked singles after elimination.
  • They're essential for intermediate to advanced puzzle solving.
  • They demonstrate advanced pattern recognition skills.

How to Spot a Hidden Triple

  1. Focus on one unit at a time.
    Pick a row, column, or box to analyze.
  2. Track where each candidate appears.
    For each number 1-9, note which cells in that unit contain it as a candidate.
  3. Look for three numbers in exactly three cells.
    Find three candidates that appear in the same three cells and nowhere else in that unit.
  4. Verify the cells have extra candidates.
    If the three cells already contain only those three candidates, it's a naked triple, not hidden.
  5. Eliminate other candidates from those cells.
    Remove all candidates except the triple from those three cells.

Hidden Triple vs Naked Triple

These complementary techniques work in opposite directions:

  • Naked triple: Three cells contain exactly three candidates between them. Eliminate those candidates from other cells in the unit.
  • Hidden triple: Three candidates appear in only three cells (which have additional candidates). Eliminate other candidates from within those three cells.

Think of it this way: naked triples eliminate outward (affecting other cells), while hidden triples eliminate inward (cleaning up the triple cells themselves).

Step-by-Step Example

Let's examine Row 8:

Row 8 candidates:

  • Col 1: {1, 4, 7}
  • Col 2: [filled with 6]
  • Col 3: {1, 3, 4, 7, 9}
  • Col 4: {1, 3, 4, 7, 8, 9}
  • Col 5: [filled with 5]
  • Col 6: {3, 8, 9}
  • Col 7: [filled with 2]
  • Col 8: {3, 8, 9}
  • Col 9: {3, 8, 9}

Analysis: Let's track where each candidate appears:

  • 1 appears in: Col 1, Col 3, Col 4
  • 3 appears in: Col 3, Col 4, Col 6, Col 8, Col 9
  • 4 appears in: Col 1, Col 3, Col 4
  • 7 appears in: Col 1, Col 3, Col 4
  • 8 appears in: Col 4, Col 6, Col 8, Col 9
  • 9 appears in: Col 3, Col 4, Col 6, Col 8, Col 9

Observation: Candidates 1, 4, and 7 appear in exactly the same three cells: Col 1, Col 3, and Col 4!

Hidden triple found: {1, 4, 7} in columns 1, 3, and 4 of Row 8.

Elimination:

  • Col 1 currently has {1, 4, 7}—already clean!
  • Col 3 has {1, 3, 4, 7, 9}—eliminate 3 and 9, leaving {1, 4, 7}
  • Col 4 has {1, 3, 4, 7, 8, 9}—eliminate 3, 8, and 9, leaving {1, 4, 7}

Result: Three cells in Row 8 now clearly contain only {1, 4, 7}, simplifying the grid and potentially revealing new patterns.

Visual Example

  • Scenario: In Column 6, you notice candidates 2, 6, and 9 appear only in rows 2, 5, and 7.
  • Before:
    • Row 2: {2, 4, 6, 8, 9}
    • Row 5: {2, 3, 6, 7, 9}
    • Row 7: {1, 2, 5, 6, 9}
  • Hidden triple identified: {2, 6, 9} are confined to these three cells.
  • After elimination:
    • Row 2: {2, 6, 9} (removed 4, 8)
    • Row 5: {2, 6, 9} (removed 3, 7)
    • Row 7: {2, 6, 9} (removed 1, 5)
  • Result: The hidden triple is now "exposed" and the cells are simplified.

Strategies for Spotting Hidden Triples Quickly

  1. Use a candidate tracking method
    Create a mental or written map of where each number appears in a unit. This makes patterns obvious.
  2. Focus on restricted candidates
    Numbers that appear in only 3-4 cells are more likely to be part of hidden triples.
  3. Look for crowded units
    Rows, columns, or boxes where most cells have 4+ candidates are prime hunting grounds.
  4. Work after simpler techniques
    Hidden triples are easier to spot once you've cleaned up obvious singles and pairs.
  5. Practice with hidden pairs first
    Master hidden pairs before focusing on triples—the logic scales up naturally.

Common Pitfalls

  • Missing the fourth cell: Make absolutely sure the three candidates appear in ONLY three cells, not four or more.
  • Confusing with naked triples: Hidden triples require elimination within the triple cells, not from other cells.
  • Incomplete elimination: Remove ALL other candidates from the three cells, not just some of them.
  • Looking across multiple units: Hidden triples must be within a single row, column, or box.
  • Forgetting to verify: Always double-check that the three candidates truly don't appear anywhere else in the unit.

Practice: Find the Hidden Triple

Try this Box 9 (bottom-right) scenario:

Box 9 cells (rows 7-9, columns 7-9):

  • Row 7, Col 7: {1, 3, 5, 6, 8}
  • Row 7, Col 8: [filled with 2]
  • Row 7, Col 9: {1, 3, 5, 6, 9}
  • Row 8, Col 7: [filled with 4]
  • Row 8, Col 8: {1, 5, 6, 7, 9}
  • Row 8, Col 9: {1, 3, 5, 7, 9}
  • Row 9, Col 7: {1, 3, 6, 8, 9}
  • Row 9, Col 8: {1, 3, 6, 7, 9}
  • Row 9, Col 9: [filled with 5]

Question: Can you find a hidden triple?

Solution: Let's track where candidates appear:

  • 1 appears in: Row 7-Col 7, Row 7-Col 9, Row 8-Col 8, Row 8-Col 9, Row 9-Col 7, Row 9-Col 8 (6 cells—too many)
  • 3 appears in: Row 7-Col 7, Row 7-Col 9, Row 8-Col 9, Row 9-Col 7, Row 9-Col 8 (5 cells—too many)
  • 6 appears in: Row 7-Col 7, Row 7-Col 9, Row 8-Col 8, Row 9-Col 7, Row 9-Col 8 (5 cells—too many)
  • 7 appears in: Row 8-Col 8, Row 8-Col 9, Row 9-Col 8 (3 cells!)
  • 8 appears in: Row 7-Col 7, Row 9-Col 7 (2 cells—not enough)
  • 9 appears in: Row 7-Col 9, Row 8-Col 8, Row 8-Col 9, Row 9-Col 7, Row 9-Col 8 (5 cells—too many)

Let's reconsider: Looking for three candidates in exactly three cells...

Actually, let me provide a clearer example where the pattern is more obvious:

Clearer Practice Example - Column 4:

Column 4 candidates:

  • Row 1: [filled]
  • Row 2: {2, 4, 5, 7, 8}
  • Row 3: {2, 5, 7, 8, 9}
  • Row 4: {1, 3, 6}
  • Row 5: {2, 4, 5, 7, 8}
  • Row 6: [filled]
  • Row 7: {1, 3, 6}
  • Row 8: {1, 3, 6, 9}
  • Row 9: [filled]

Track candidates:

  • 1 appears in: Row 4, Row 7, Row 8
  • 3 appears in: Row 4, Row 7, Row 8
  • 6 appears in: Row 4, Row 7, Row 8

Hidden triple found: {1, 3, 6} in rows 4, 7, and 8 of Column 4!

Action:

  • Row 4 already has {1, 3, 6}—perfect!
  • Row 7 already has {1, 3, 6}—perfect!
  • Row 8 has {1, 3, 6, 9}—eliminate 9, leaving {1, 3, 6}

Why Hidden Triples Set the Stage

Hidden triples extend the logic of hidden pairs and complete your intermediate toolkit:

  • They prepare you for hidden quads (four candidates in four cells)
  • They improve your ability to track candidate distribution systematically
  • They often work together with naked triples in the same unit
  • They're essential for solving challenging intermediate and advanced puzzles
  • They develop the pattern recognition needed for expert techniques

Quick Recap

Technique Pattern Elimination Direction Difficulty
Naked Triple 3 cells, 3 candidates Outward (other cells) Intermediate
Hidden Triple 3 candidates in 3 cells only Inward (triple cells) Intermediate
Hidden Pair 2 candidates in 2 cells only Inward (pair cells) Intermediate
Hidden Quad 4 candidates in 4 cells only Inward (quad cells) Advanced

Final Thought

When a unit is crowded with candidates, ask yourself: are there three numbers confined to just three cells? Hidden triples might be lurking, waiting to simplify your grid.

Frequently Asked Questions

What is a Hidden Triple in Sudoku?

A hidden triple occurs when three candidates appear in only three cells within a unit (row, column, or box), but those cells also contain other candidates that mask the triple. Once identified, you can eliminate all other candidates from those three cells, leaving only the hidden triple.

How do I spot a Hidden Triple?

To spot hidden triples: 1) Focus on one unit at a time, 2) For each candidate number, note which cells contain it, 3) Look for three numbers that appear in exactly the same three cells (and nowhere else in that unit), 4) Eliminate all other candidates from those three cells, leaving only the triple.

What's the difference between Hidden Triples and Naked Triples?

Naked triples have three cells with only three candidates visible between them, and you eliminate outward from those cells. Hidden triples have three candidates that appear in only three cells (which have extra candidates), and you eliminate inward within those cells. Naked triples eliminate from other cells; hidden triples clean up the triple cells themselves.

Why are Hidden Triples important?

Hidden triples are important because they unlock progress in crowded units where other techniques fail, they simplify complex cells with many candidates, they often reveal naked singles after elimination, and they're essential for intermediate to advanced puzzle solving.

When should I look for Hidden Triples?

Look for hidden triples after exhausting simpler techniques like naked and hidden singles, naked and hidden pairs. They're especially useful in units where many cells have 4-6 candidates each. Systematically check each unit, tracking which three numbers appear in exactly three cells.

Ready to advance? Check out our complete strategy guide for more techniques.

Related Strategies

Once you've mastered hidden triples, these techniques build naturally:

  • Hidden Pairs - The foundation for hidden triples
  • Naked Triples - The complementary technique
  • Hidden Quads - Four-cell extension of this logic (coming soon)
  • Naked Quads - Four-cell version of naked triples (coming soon)
  • X-Wing - Advanced pattern recognition technique

Practice Hidden Triples

Next up: Try Naked Quads or XYZ-Wing to advance to expert techniques (coming soon).

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